Number Strings

A string is a chain of related problems, served one at a time: $2^3\times 2^2$, then $2^5\times 2^0$, then $2^7\times 2^{-2}$, then $2^{1/2}\times 2^{1/2}$. Each answer is a stepping stone to the next — and in Grade 11, the strings are quietly rebuilding the exponent laws from scratch.

How we play

1. One problem at a time. Solve it in your head; thumb when ready.
2. A few people defend their methods; each goes on the board.
3. Before computing the next from scratch, ask: what can I reuse?

The string, walked

• ”$2^3\times 2^2$: eight fours — no wait, eight times four, 32. Which is $2^5$. The exponents added.”
• ”$2^5\times 2^0=32$ only works if $2^0=1$. Not zero — one.”
• ”$2^7\times 2^{-2}$: if the law still holds it is $2^5=32$ again, so a negative exponent must divide.”
• “$2^{1/2}\times 2^{1/2}=2^1$. Whatever $2^{1/2}$ is, it multiplies by itself to make 2. It is $\sqrt{2}$.”

Nobody was taught a rule that day. The string cornered each one, and Exponent Laws just wrote down what the room had proven.

One variation

Run the string with money: $200\times 1.05$, then $200\times 1.05^2$, then $200\times 1.05^{10}$ — a balance compounding before anyone has said the word. The last step of the original string is the front door of Rational Exponents.

Lazy, in the best way

Mathematicians refuse to compute what they can deduce. If a problem feels brand new, look back along the string — it rarely is.